L(s) = 1 | + 2-s − 4-s + 3·5-s + 2·7-s − 3·8-s − 3·9-s + 3·10-s + 11-s + 2·14-s − 16-s − 8·17-s − 3·18-s − 5·19-s − 3·20-s + 22-s + 7·23-s + 4·25-s − 2·28-s − 5·29-s + 6·31-s + 5·32-s − 8·34-s + 6·35-s + 3·36-s + 2·37-s − 5·38-s − 9·40-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1/2·4-s + 1.34·5-s + 0.755·7-s − 1.06·8-s − 9-s + 0.948·10-s + 0.301·11-s + 0.534·14-s − 1/4·16-s − 1.94·17-s − 0.707·18-s − 1.14·19-s − 0.670·20-s + 0.213·22-s + 1.45·23-s + 4/5·25-s − 0.377·28-s − 0.928·29-s + 1.07·31-s + 0.883·32-s − 1.37·34-s + 1.01·35-s + 1/2·36-s + 0.328·37-s − 0.811·38-s − 1.42·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.411025916\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.411025916\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 109 | \( 1 - T \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.51696514179328766601443682280, −13.16351918145631944214330863164, −11.68350949166308979033668111954, −10.66560530782850073835534538181, −9.150313873860531819864186711154, −8.647170498136538712905651969481, −6.52096433321379207755161651531, −5.53602205460949346898381900178, −4.44940245252522610753026878032, −2.44852333863076941577077311566,
2.44852333863076941577077311566, 4.44940245252522610753026878032, 5.53602205460949346898381900178, 6.52096433321379207755161651531, 8.647170498136538712905651969481, 9.150313873860531819864186711154, 10.66560530782850073835534538181, 11.68350949166308979033668111954, 13.16351918145631944214330863164, 13.51696514179328766601443682280